This is not quite true, Gödel's incompleteness theorems can luckily be formalized in extremely weak fragments of Peano arithmetic such as primitive recursive ar

This is not quite true, Gödel's incompleteness theorems can luckily be formalized in extremely weak fragments of Peano arithmetic such as primitive recursive arithmetic (PRA) with its very limited induction principle. :-) The only position on the philosophy of mathematics I know which does not accept PRA is ultrafinitism.